Maps preserving strong 2-Jordan product on some algebras

Abstract

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

Strong 2-Jordan product preserving maps on operator algebras

“…It is well known that standard operator algebras are prime. Hence, by Corollary 2.2, the following result is obvious, which generalizes Theorem 2.1 in [12].…”

“…It is shown that, under some mild conditions, f satisfies {f (a), f (e)} 2 = {a, e} 2 for all a ∈ R and e ∈ {e 1 , 1 − e 1 , 1} if and only if f (1) is in the center of R with f (1) 3 = 1 and f (a) = f (1)a holds for all a ∈ R (Theorem 2.1). As applications, such maps on prime rings, standard operator algebras and von Neumann algebras are characterized, respectively (Corollaries 2.2-2.4), which generalize the corresponding results in [10,12].…”

“…Assume that R is a unital ring containing a nontrivial idempotent and f : R → R is a surjective map. Wang and Qi [10] showed that, under some mild conditions, f is strong k-Jordan product preserving if and only if there exists λ ∈ Z(R) (the center of R) with λ k+1 = 1 such that f (x) = λx holds for all x ∈ R. Taghavi, Kolivand and Rohi in [11] proved that, if A is a unital algebra containing a nontrivial idempotent e 1 , then a surjective map f : A → A satisfies {f (a), f (e)} 1 = {a, e} 1 for all a ∈ A and e ∈ {e 1 , 1 − e 1 } if and only if f (1) is in the center of A, f (1) 2 = 1 and f (a) = f (1)a for all a ∈ A; in [12] gave a concrete form of strong 2-Jordan product preserving surjective maps on standard operator algebras, and particularly, showed that, if a surjective map Φ : M → M (here, M is a properly infinite von Neumann algebra) satisfies {Φ(A), Φ(P )} 2 = {A, P } 2 for all A ∈ M and all idempotents P ∈ M, then Φ(A) = Φ(I)A for all A ∈ M.…”